Necessary and sufficient conditions for the existence of an orthogonal $\ast$-basis of symmetry classes of tensors associated to nonabelian groups of order $pq$ are provided by using vanishing sums of roots of unity.

We show that the complexity of the Lie module Lie(n) in characteristic p is bounded above by m, where pm is the largest p-power dividing n, and, if n is not a p-power, is equal to the maximum of the complexities of Lie(pi) for 1≤i≤m.

We present, given an odd integer d, a decomposition of the multiset of bar lengths of a bar partition λ as the union of two multisets, one consisting of the bar lengths in its d-core partition cd(λ) and the other consisting of modified bar lengths in its d-quotient partition. In particular, we obtain that the multiset of bar lengths in cd(λ) is a sub-multiset of the multiset of bar lengths in λ. Also, we obtain a relative bar formula for the degrees of spin characters of the Schur extensions of . The proof involves a recent similar result for partitions, proved by Bessenrodt and the authors.

Let K be a field of characteristic p. The permutation modules associated to partitions of n, usually denoted as Mλ, play a central role not only for symmetric groups but also for general linear groups, via Schur algebras. The indecomposable direct summands of these Mλ were parametrized by James; they are now known as Young modules; and Klyachko and Grabmeier developed a ‘Green correspondence’ for Young modules. The original parametrization used Schur algebras; and James remarked that he did not know a proof using only the representation theory of symmetric groups. We will give such proof, and we will at the same time also prove the correspondence result, by using only the Brauer construction, which is valid for arbitrary finite groups.